JoLY — The Associative Algebra applicable to Hyperspace, 109 



From this it is evident that 

 (|> (^■l4) = V2 (^■^^^l - ii<i>i%) = (ffi + ffz) iA = 2m {x^^ + x^^) i^, 

 or the area vectors of planes containing two principal axes of inertia 

 are the solutions of $0 = cB, where <? is a root of $. 



46, It is easy to investigate the conditions for the steady motion 

 of a body under no forces. In general (compare Art. 10), 



Bt^Q, = <i>n + Fsfl^o = r, 



where r is the couple referred to the centre of mass. If V is zero, and 

 if O vanishes also, the equation of motion becomes FsO^O = 0. 



This is of course satisfied when O = ainj-iio ; but it is also satisfied 

 when Q, is in the canonical form for these units, or when 



O = aioiiiz + a^iizii + . . . 



So, if an impulsive couple acts on a body which has been so j)laced 

 that the components of the couple are all contained in principal planes 

 of the body which are hyper-perpendicular to one another, the body 

 will permanently rotate with constant angular velocities in each of 

 these principal planes. 



47. The linear function $ which, by operating on the generalized 

 angular velocity, produces the generalized angular momentum, is by 

 no means the most general of the type to which it belongs. Its roots, 

 ^m{m - 1) in number, are the sums of pairs of the m roots of a self- 

 conjugate linear vector function of the ordinary type ; its axes, more- 

 over, are pure area vectors. 



Regarded as derived from an ordinary linear vector function, it 

 belongs to an extensive type of functions of which a few examples are 

 now given. 



Consider the condition that a function /(-4) of a quadratic in the 

 units {A) and itself quadratic in the units, should be expressible in 

 the form 



f{A) = s/FsV = %ro{e,xeo^ix + 6,\e,ix), 



where 61, Oo, 63, and ^4 are ordinary linear vector functions. 

 It is necessary for all vectors A and fi that 



/ K\ix = r^ie^xe^fji + e^xOifi) = - v^ieifjid^x + ^sM^) = -fr^ixX. 



This requires Oo = 0^ = <^, and 6i = 61 = 6, 



or else 0^ = 0^ = 0, and 63 = $i = eft; 



so the two admissible types 



/i V^XfjL = Fo {OX^ix + <i>Xe^ji), and /^ V^Xijl = V^ {6X0 fx + (^X<^/x) 

 are found. 



